The quantum circuit shown in Fig. REF  completed a post-selection Toffoli gate with a success probability of \(P_\text{Toff}=\frac{1}{2}\times \frac{1+\alpha _2^2}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}= \frac{1+\alpha _2^2}{64}\) .
Alternately, the success probability can be further improved to \(\frac{1}{2}\times \frac{1+\alpha _2^2}{2}\times \frac{2-\alpha _2^2}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}= \frac{(1+\alpha _2^2)(2-\alpha _2^2)}{64}\)  by recycling the photons emitted from spatial mode \(5^{\prime \prime }\)  into the PBS\(_5\)  because the state has the same form as the state shown in Eq. (REF ).
In Fig. REF , we can see that the minimum \(P_\text{Toff}\)  without recycling is approximately \(\frac{1}{64}\) , the maximum \(P_\text{Toff}\)  without recycling reaches approximately \(\frac{1}{32}\) , and the average \(P_\text{Toff}\)  without recycling is \(\frac{1}{2} \int _{-1}^{1}\frac{1+\alpha _2^2}{64}d\alpha _2=\frac{1}{48}\) . In contrast, with recycling one, the minimum \(P_\text{Toff}\)  is approximately \(\frac{1}{32}\) , the maximum \(P_\text{Toff}\)  can reach \(\frac{9}{256}\) , and the average \(P_\text{Toff}\)  is \(\frac{1}{30}\) . The success probability of our Toffoli gate is much higher than previous works [1]}, [2]}, [3]}, [4]}.
<TABLE><TABLE>